nih public access serkis c. isikbay jie chen angle orthod · a substantial second order tipping...

An analytical approach to 3D orthodontic load systems

Thomas R. Katonaa, Serkis C. Isikbayb, and Jie Chenc

aAssociate Professor, Department of Orthodontics and Oral Facial Genetics, Indiana UniversitySchool of Dentistry, and Department of Mechanical Engineering, Purdue University School ofEngineering and Technology, Indianapolis, Ind

bPrivate practice, Speedway, Ind

cProfessor, Department of Mechanical Engineering, Purdue University School of Engineering andTechnology, and Department of Orthodontics and Oral Facial Genetics, Indiana University Schoolof Dentistry, Indianapolis, Ind

Abstract

Objective—To present and demonstrate a pseudo three-dimensional (3D) analytical approach for

the characterization of orthodontic load (force and moment) systems.

Materials and Methods—Previously measured 3D load systems were evaluated and compared

using the traditional two-dimensional (2D) plane approach and the newly proposed vector method.

Results—Although both methods demonstrated that the loop designs were not ideal for

translatory space closure, they did so for entirely different and conflicting reasons.

Conclusions—The traditional 2D approach to the analysis of 3D load systems is flawed, but the

established 2D orthodontic concepts can be substantially preserved and adapted to 3D with the use

of a modified coordinate system that is aligned with the desired tooth translation.

Keywords

Three-dimensional; Orthodontic force systems; Biomechanics; T-loop archwire; Moment-to-forceratio

INTRODUCTION

Orthodontic load (force and moment) systems are traditionally quantified as plane two-

dimensional (2D) arrangements.1–7 Three-dimensional (3D) systems have typically been

viewed as combinations of three such 2D planes oriented in each tooth’s facial/buccal,

incisal/occlusal and mesiodistal directions. But 3D engenders more complex interactions.7–9

Therefore, a purpose of this paper is to show that the loads in the 2D planes are not

independent, and therefore, their actions should not be uncoupled and treated separately

because that leads to errors in the prediction of orthodontic displacements. The second

© 2014 by The EH Angle Education and Research Foundation, Inc.

Corresponding author: Dr Thomas R. Katona, Indiana University School of Dentistry, IUPUI, 1121 W Michigan St, Indianapolis, IN46202 ([email protected]).

NIH Public AccessAuthor ManuscriptAngle Orthod. Author manuscript; available in PMC 2015 March 01.

Published in final edited form as:Angle Orthod. 2014 September ; 84(5): 830–838. doi:10.2319/092513-702.1.

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purpose is to present a pseudo-3D analytical approach that substantially preserves the

concepts and nomenclature of the accepted 2D methods. Finally, the third purpose is to

show experimental results used to demonstrate and contrast the analyses.

Consider, as viewed in the traditional 2D analytical approach, the translations of a maxillary

left lateral incisor and canine into an extraction space. As observed in the occlusal plane

(Figure 1), there are three pertinent coordinate systems. X-Y is the global system and there

are two x-y systems associated with the respective buccal and distal sides of each tooth. Z

and z are in the superior/apical direction. Thus, X-Y and the two x-y define the occlusal

plane. (X-Z defines the midsagittal plane and Y-Z defines a frontal plane.)

The most general 3D load system that a loop can apply to a bracket consists of three force

and three moment components (Figure 2a). In the typical traditional 2D approach, this 3D

load system is independently analyzed in tooth-relative orthogonal planes formed by the x-y,

y-z, and x-z axes of each tooth (Figure 1). In the y-z plane (Figures 1b and 2a), the force

components are Fy and Fz, and the moment component, ±Mx, acts in the buccal/palatal

direction (according to the right-hand-rule [RHR] convention) to compensate for second

order rotation. Third order tipping takes place in the x-z plane with Fx, Fz, and My. And, in

the x-y (occlusal) plane, the forces are Fx and Fy, and Mz is associated with mesial-in/distal-

out rotation. Thus, the most general 2D model contains only two of the three force

components, and only one of the three moment components.

To associate a load system on the bracket with the resulting tooth displacement, the tooth’s

center of resistance (CRes) is taken into account. CRes is an imaginary point fixed relative

to the tooth, that, if a force were applied to it, the tooth would undergo pure translation in the

direction of that force. (An alternative definition is that the tooth undergoes pure rotation

about CRes if only the moment of a couple is applied to the tooth.) The location of CRes is

determined by the shapes and the mechanical properties of the root, socket, and periodontal

ligament matrix and fibers.10–12 In a single rooted tooth, it is somewhere within the middle

third of the root.

Thus, viewed in 2D, for the space-closing translation of the lateral incisor, a force in its

distal direction (Fy) would have to be applied to its CRes (Figure 3a). At the bracket, the

equivalent load system consists of the same Fy plus a moment, −Mx (Figure 3b,c). The

magnitude of Mx should be (Fy) (dz ) where dz, the moment arm, is the distance between

CRes and the line-of-action (LOA) of Fy (Figure 3a). Thus, for distal translation of the tooth,

the moment-to-force ratio (M/F) that has to be applied to the bracket by the appliance is

Mx/Fy (= dz). In effect, −Mx serves to eliminate the second order rotation caused by moving

Fy from CRes to the bracket. Second order gable bends are often used to augment

insufficient loop-generated Mx/Fy values.

In total, there are nine M/F ratio permutations, but because forces cannot generate moments

in their own directions (as defined by the RHR), the crossed-out quantities have no meaning.

In the customary 2D segmental view of space closure (Figures 1 through 6), the emphasis is

on Fy because it is the force component in the desired tooth movement direction and because

it has the longest (8–10 mm) moment arm (dz) relative to CRes. That combination produces

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a substantial second order tipping moment about CRes so its counteraction (with Mx) has

been the traditional focus of loop design and gable bend studies aimed at increasing Mx/Fy to

the necessary 8–10 mm magnitude.13 (The generic “moment-to-force ratio” in the literature

usually refers implicitly to Mx/Fy.)

An analytical method should be applicable to the most general loading (Figure 2a).

Therefore, the effects of all forces, moments, and M/F ratios must be taken into account. In

the plane discussed above (Figure 3), another ±Mx component can be produced by the

intrusive/extrusive force component (±Fz) on the bracket (Figure 4), which must be opposed

by an applied Mx/Fz that is equal to dy, the distance (ie, the moment arm) between CRes and

the LOA of Fz. However, in actuality, in this plane, the Mx that is applied to the tooth by the

appliance must negate the net Mx on the tooth that is produced by the combined actions of

Fy and Fz (Figures 3 and 4). Perhaps because dy is relatively small when an upright tooth is

translated, and because Fz is also relatively small, the contribution of Fz is usually ignored.

In the occlusal plane, the same Fy that produces an Mx (Figure 3) also produces an Mz

(mesial-out, distal-in rotation) (Figure 5a). That moment must be counteracted by a −Mz (=

Fydx) that is generated by the loop. Another Mz is produced by Fx, and it must be negated by

an Mz/Fx that is equal to dy (Figure 5b). But, analogous to the y-z plane (Figures 3 and 4),

the appliance-applied M/F in this plane must counteract the net Mz produced by Fy (Figure

5a) and Fx (Figure 5b). And finally, My moment (“torque”) in the x-z plane is produced by

Fx and Fz due to their respective moment arms, dz and dx (Figure 6).

Thus, a force component can produce moments in the two directions that are perpendicular

to it, or a moment component can be created by force components in the other two directions

(Figure 7). Therefore, the separation of the complex 3D phenomena into three simpler 2D

models can create problems because, in reality, those 2D models are not independent of each

other. Change in one plane, a gable bend for example, has the potential to impart changes in

the other two planes.

The above discussions have been limited to the traditional 2D analysis, and therefore, to

tooth translations and rotations relative to their individual x-y-z (buccal-distal-apical)

coordinate systems. In 3D scenarios involving the arch, this can present severe shortcomings

because the orientation of x-y-z for each tooth is different relative to each other and relative

to the global system (Figure 1a). Therefore, unlike the components shown in Figure 2a, a

distal force component on the lateral incisor, +Fy, is not in the opposite direction to a mesial

force component on the canine, −Fy. Thus, concepts based on Figure 2a are applicable to the

unrealistically distorted arches in Figure 2b or c.

MATERIALS AND METHODS

It is proposed that for the purposes of analyzing and characterizing 3D orthodontic load

systems, the analysis be conducted in coordinate systems that are more closely aligned with

the approximated directions of desired translations, the y′ in Figure 8. By so doing and

slightly modifying the nomenclature of the above described conventional x-y-z based 2D

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systems, mainstream orthodontic concepts and approaches can be salvaged and adapted to

3D.

Using the analyses of load systems in a simulated translatory space closure, the proposed

and traditional methods are compared. Zero-, 1-, and 2-mm activation results from 2 of the

16 loop designs, configurations no. 13 and no. 8 in the previous publication,7 are used as

exemplars.

First, it is estimated that the incisor and canine x-y axes are rotated 30° and 65°,

respectively, relative to X-Y (Figure 1a). Then, with transformation equations, 14 the x-y

system load components are projected onto the X-Y global system and defined on each tooth

as FG and MG (Figure 9). Then, it can be ascertained if the FG forces act along their

respective desired (y′) directions, estimated as approximately 50° and 35°. If deemed

acceptable, the MG must have components, M⊥, that are perpendicular to FG, with the

appropriate senses and M⊥/FG ratios to prevent unwanted tipping (Figure 9b).

RESULTS

For the first loop design, the load systems on the brackets are presented traditionally in each

tooth’s respective x-y-z system (Figure 10). The same data are shown according to the

proposed method in Figure 11. The other loop design results are depicted in Figure 12.

DISCUSSION

A purpose of this project is to illustrate the differences, advantages, and deficiencies of the

two approaches when applied to the identical experimental measurements. The traditional

presentation of data (Figure 10) presents critical pitfalls when extrapolated to 3D mainly

because the desired tooth displacements rarely coincide with the x-y-z coordinate system of a

tooth and because the two x-y-z systems are not aligned. Thus, 3D interactions are better

handled vectorially, as proposed.

An acceptable load system should produce the intended tooth displacements (in this case,

translation into the extraction space) with minimal side effects. According to the

conventional 2D approach, for space closing translation of the incisor, the following

conditions are required: the force should be distally directed (Fy > 0) and the moment should

be in the palatal direction (Mx < 0) (Figure 2), with Mx/Fy between −8 and −10 mm.

Concomitantly, on the canine, the force should be mesial (Fy < 0), and the moment should

be buccally directed (Mx > 0), and Mx/Fy should be between −8 and −10 mm. With the first

loop configuration, at 1- and 2-mm activation, −10 < Mx/Fy < −8 is produced on the canine

(Figure 10d). The Fy < 0 and Mx > 0 requirements are also met (Figure 10c,d). However, this

loop wreaks havoc on the incisor, with Mx/Fy equal to −62.0 and −109.0 mm, respectively,

at the 1- and 2-mm activations. When viewed in the occlusal plane (Figure 11), the

inappropriate force directions are immediately obvious.

Thus, 3D analysis can be manageably performed for each tooth by vectorially adding its Fx

and Fy, and its Mx and My, and expressing the sums in the global X-Y system as FG and MG

for each tooth (Figures 9, 11, and 12). Then it becomes possible to apply the following

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requirements, in any sequence, for a suitable translatory space closing load system (Figure

9):

• R1: The force vector, FG, must be in the required translatory direction;

• R2: The associated moment vector (MG) must have a component (M⊥) in the

appropriate direction, ie, perpendicular to FG and with the correct sense as defined

by the RHR;

• R3: The required M⊥/FG ratio must be met;

• R4: The magnitude of FG must be within some suitably defined range;

• R5: Load components Fz and Mz must be within defined acceptable ranges; and

• R6: M||, the component of MG that is parallel to FG (Figure 9b), hence

perpendicular to M⊥, must be sufficiently small.

If any requirement is violated on either tooth, then all other conditions are beside the point.

(These requirements are entirely independent of the orthodontic appliance.) The evaluation

process is illustrated as follows:

On the canine, the second spring configuration produces acceptable force directions at 1 and

2 mm of activation (Figure 12b), thus meeting condition R1. Their respective M⊥/FG are

−11.1 and −9.6 mm. (The traditional Mx/Fy are −12.1 and −19.1 mm, respectively.) Thus, at

1-mm activation, the force direction is optimal, but the M⊥/FG ratio is out of range, thereby

violating condition R3. Although the 2-mm activation satisfies requirements R1, R2, and

R3, R6 is questionable because MG has a relatively large M|| component that would cause a

substantial modified third order rotation. (“Modified” because the crown tips not only in the

traditional palatal direction, but also toward the distal.) But, assuming for the sake of

argument that R6 is acceptable, condition R5 must be examined. The roles of Fz (intrusion/

extrusion) and Mz (mesial-in/distal-out rotation) in this analysis are exactly as they are in the

traditional approach. Note the concomitant gross violations of R1 on the incisor in Figure

12a.

The proposed approach eliminates the use of FG whose directions are deemed unacceptable.

This simplifies the analysis because there is no analogous ±Fx′ force component. Further

simplification of the 3D analysis is possible. Figure 8b shows the assumption of collinear

paths of the two teeth along the y″-axis. So, instead of performing the traditional individual

analyses in the x-y-z coordinate systems of each tooth (Figure 1a), analyses can be done in

their x′-y′-z systems. This replicates the traditional approach depicted in Figures 2 through 6.

Thus, in effect, an appropriate FG is analogous to Fy in the traditional approach and M⊥ is

analogous to Mx. The magnitudes and directions of Fx″ and M|| (which is analogous to My)

indicate the level of potential unwanted side effects, another advantage of this approach.

The requirements for translation, the illustrative example, are listed above as R1 to R6. But

the proposed approach is adaptable to any tooth displacement if the requirements are

appropriately modified. In all cases, the FG must be in the direction of desired translation

and the M/F ratios must be specified for the desired rotations and tipping, just as with the

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conventional approach. The essential difference is that the analyses are performed relative to

the y′, not y.

The analyses illustrate the intricate nonintuitive confounding effects of 3D. The well-defined

concepts and principles of 2D mechanics (Figures 2 through 6) cannot be directly

extrapolated to curved-arch 3D, and the two analytical methods attribute loop failure to

different reasons. (For the purposes of comparing the analytical approaches, loop design is

irrelevant.) Although far from a panacea, the proposed displacement- relative analysis

allows for established 2D concepts to be applied in 3D.

CONCLUSIONS

• The traditional 2D approach to the analysis of 3D load systems is flawed.

• Traditional 2D orthodontic concepts can be adapted to a pseudo-3D analysis with

the use of a modified coordinate system that is aligned with the desired tooth

translation direction.

Acknowledgments

The study was partially supported by grants, NIH-NIDCR R41-DE017025 and R01 DE018668.

References

1. Chen J, Bulucea I, Katona TR, Ofner S. Complete orthodontic load systems on teeth in a continuousfull archwire: the role of triangular loop position. Am J Orthod Dentofacial Orthop. 2007; 132:143,e141–148. [PubMed: 17693357]

2. Chen J, Markham DL, Katona TR. Effects of T-loop geometry on its forces and moments. AngleOrthod. 2000; 70:48–51. [PubMed: 10730675]

3. Gregg, J.; Chen, J. The effect of wire fixation methods on the measured loading systems of a T-looporthodontics spring. Paper presented at: AAO Annual Meeting; Month DD, 1997; Philadelphia, Pa.

4. Katona TR, Le YP, Chen J. The effects of first- and second-order gable bends on forces andmoments generated by triangular loops. Am J Orthod Dentofacial Orthop. 2006; 129:54–59.[PubMed: 16443479]

5. Lisniewska-Machorowska B, Cannon J, Williams S, Bantleon HP. Evaluation of force systems froma “free-end” force system. Am J Orthod Dentofacial Orthop. 2008; 133:791, e1–10. [PubMed:18538238]

6. Raboud D, Faulkner G, Lipsett B, Haberstock D. Three-dimensional force systems from verticallyactivated orthodontic loops. Am J Orthod Dentofacial Orthop. 2001; 119:21–29. [PubMed:11174536]

7. Katona TR, Isikbay SC, Chen J. Effects of first- and second-order gable bends on the orthodonticload systems produced by T-loop archwires. Angle Orthod. 2013 Aug 29. Epub ahead of print.

8. Badawi HM, Toogood RW, Carey JP, Heo G, Major PW. Three-dimensional orthodontic forcemeasurements. Am J Orthod Dentofacial Orthop. 2009; 136:518–528. [PubMed: 19815153]

9. Chen J, Isikbay SC, Brizendine EJ. Quantification of three-dimensional orthodontic force systems ofT-loop archwires. Angle Orthod. 2010; 80:754–758.

10. Qian H, Chen J, Katona TR. The influence of PDL principal fibers in a 3-dimensional analysis oforthodontic tooth movement. Am J Orthod Dentofacial Orthop. 2001; 120:272–279. [PubMed:11552126]

11. Meyer BN, Chen J, Katona TR. Does the center of resistance depend on the direction of toothmovement? Am J Orthod Dentofacial Orthop. 2010; 137:354–361. [PubMed: 20197172]

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12. Viecilli RF, Budiman A, Burstone CJ. Axes of resistance for tooth movement: does the center ofresistance exist in 3-dimensional space? Am J Orthod Dentofacial Orthop. 2013; 143:163–172.[PubMed: 23374922]

13. Proffit, WR.; Fields, HW.; Sarver, DM. Contemporary Orthodontics. 4. Chicago, Ill: CV Mosby;2007.

14. Beer, FP.; Johnston, ER.; Eisenberg, ER. Vector Mechanics for Engineers - Statics. 8. Boston,Mass: McGraw-Hill; 2007.

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Figure 1.(a) Occlusal view of the simulated clinical case showing the global (X-Y) and the estimated

local (x-y) coordinate systems on the maxillary lateral incisor (approximately 30°) and the

canine (approximately 65°) brackets. (b) The x-y-z axes of the lateral incisor bracket.

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Figure 2.(a) Schematic of complete generic load system acting on a maxillary left lateral incisor

bracket. ±Fx, ±Fy, and ±Fz force components act in the buccal/palatal, distal/mesial, and

apical/incisal directions, respectively. The directions of the moment vector components

(±Mx, ±My, and ±Mz) are defined by the RHR convention—the thumb of the right hand

points in the direction of the moment (open) vector arrow, and the fingers indicate the

direction of rotation. As an example, −Mz would produce a mesial-in distal-out rotation. (b)

The distorted arch corresponds to the depictions in Figure 2a and in Figure 1a. (c) Distorted

arch based on canine x-y data.

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Figure 3.Load systems required for distal translation applied at (a) CRes or at (b and c) the bracket. b

and c use, respectively, the RHR and the more (dentally) conventional depictions.

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Figure 4.For translation, Mx also counters the second order rotation produced by Fz.

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Figure 5.Mz = −Fydx + Fxdy to prevent rotations produced by (a) Fy and (b) Fx.

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Figure 6.My is applied to prevent third order rotation produced by Fx and Fz in the x-z plane. My =

Fxdz + Fzdx.

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Figure 7.Each force component can generate two moment components. Each moment component can

be generated by two force components.

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Figure 8.(a) Instead of traditional analyses in the two x-y coordinate systems (Figures 1 through 6), it

is proposed that analyses be performed in the two x′-y′ systems in which the y′-axes are

more closely aligned (approximately 50° instead of approximately 30° for the incisor and

approximately 35° instead of approximately 65° for the premolar) with the direction of

desired tooth translations. (b) Analogous to Figure 2a, the assumption that the two y′-axes

are collinear yields the y″-axis. Its angulation, 43°, is the average of the 35° and the 50°.

(Alternatively, the 43° approximates the line joining the two brackets.) With this

approximation, the gross arch distortion (Figure 2b) is avoided, and the traditional widely

accepted concepts associated with Figure 2a are more acceptable.

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Figure 9.(a) For the desired tooth translations, each FG must approximate its y′-axis. With an

appropriate FG, as on the canine for example, to counteract its tipping action, (b) there must

be a component of MG, M⊥, which is perpendicular to the force. M|| is the component of

MG that is parallel to FG.

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Figure 10.Traditional presentation of force and moment components on the brackets produced by the

first loop design. The results for 0, 1, and 2 mm of activations on the (a and b) incisor and (d

and e) canine.

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Figure 11.The same data as in Figure 10, but as vectors (FG and MG) projected onto the occlusal plane

for the (a) incisor and (b) canine. The 0, 1, and 2 indicate activations in mm.

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Figure 12.As in Figure 11 but for the second loop design.

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Table 1

Mx/Fx My/Fx Mz/Fx

Mx/Fy My/Fy Mz/Fy

Mx/Fz My/Fz Mz/Fz


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